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I added some simpler motivation in terms of the basic example to the beginning of distributive law.
Yes, nice. Rings make an obvious choice to illustrate a distributive law between monads. Then we’re introduced to the scary thought that there’s a whole 2-category of such laws. Could this be illustrated by a surprising example or two of a distributive law, perhaps something one would never have thought to be such a law?
one would never have thought to be such a law
The level of difficulty in fullfilling this task depends on who is "one" who never thought...
There’s definitely a gap between the first two sections of this page which someone who has some time on their hands should fill. (-:
a surprising example
I for one was quite tickled to find out that factorization systems are distributive laws between categories taken as monads in Span(Set). That might be a good non-obvious example.
Good. I’ve put that example in.
It’s not quite true as stated, though – ordinary distributive laws in Span(Set) only give you “strict” factorization systems. I corrected it.
I keep pointing to “distributive law” whenever I mention something distributive. Now looking back at the entry to see what it actually says, I notice that many of the common “distributivities” were not really mentioned. I added a quick pointer to distributive category and to tensor products distributing over direct sums. But this needs to be better incorporated into the entry, eventually.
added these pointers:
Stephen Brookes, Kathryn Van Stone, Monads and Comonads in Intensional Semantics (1993) [dtic:ADA266522, pdf]
John Power, Hiroshi Watanabe, Combining a monad and a comonad, Theoretical Computer Science 280 1–2 (2002) 137-162 [doi:10.1016/S0304-3975(01)00024-X]
added cross-link to the new entry weak distributive law and to pseudo-distributive law
spelled out (here) the mixed distributivity law of a monad over a comonad (from Brookes & Van Stone 1993 Def. 3)
distributivity law of a monad over a comonad
I would just like to warn that the prepositions etc. in phrases for distributive laws are not standardized. One hear law from monad to comonad, from comonad to monad, between monad and comonad, between comonad and monad etc. and I never found a way to logically remember any of those. I mean for the mixed law (entwining structure) there is no problem – there is just one, but when there are two monads or two comonads, there are two different laws – then it does matter which one is first and which the second, but which order corresponds to whose usage of grammar is totally random.
I would just like to warn that the prepositions etc. in phrases for distributive laws are not standardized.
There is (what ought to be) a standard, but I have heard people get it wrong. The phrasing should align with “multiplication distributes over addition”, which means that we have a transformation ${\times} \circ {+} \Rightarrow {+} \circ {\times}$. I.e. $S$ distributes over $T$ when we have a transformation $ST \Rightarrow TS$. Using any other terminology or convention is at best ambiguous and at worst misleading.
I notice the nLab page is not entirely consistent about this and talks about “distributives law from one thing to another”. If no-one has any objections, I’ll clean this up.
In the mixed case I have been following Brookes & Van Stone 1993, Def. 3 who speak the other way around p. 12.
But it’s opposite to what one should say, yes.
Incidentally, Brookes & Van Stone is the only reference for the mixed case that I am aware of (in my ignorance). Maybe we can find another reference for the mixed case which has the terminology straight.
Maybe we can find another reference for the mixed case which has the terminology straight.
Power and Wanatabe use the expected terminology (opposite to that of Brookes and Van Stone) in Combining a monad and a comonad (e.g. see the introduction, or Definition 6.1 on p. 153).
Urs 18, I repeat – in the case of a monad and comonad there are NO two versions just one, so the direction matters only when there are two monads or two comonads.
The notion is also self-dual in the case of a monad and comonad, as Brzezinski put it, if you reverse all the arrows (then monads exachange for comonads, algebras with coalgebras etc.) you get the same diagram.
varkor, thanks,
I have adjusted the wording here and added pointer to Power & Watanabe.
Unfortunately, Power & Watanabe don’t seem to explicitly state the distributivity of a comonad over a monad(?) – their Def. 6.1 only states the distributivity of a monad over a comonad. Not that it’s a big deal, but it means one can’t quite cite them for it.
Also, they attribute the notion of comonad distributing over a monad to Brookes & Geva 1991 (pdf, instead of Brookes & Van Stone) but I don’t spot any discussion of distributivity in there(?)
Mixed distributive laws are stated and used in 1970s also in
Now I see that Brookes and Van Stone use inverse-looking map, but it is doubtful to call it a distributive law. I mean the following.
Look, if you have two monads, $T$, $S$, then a distributive law $TS \to ST$ lifts $S$ to a monad on $C^T$ and the opposite law $ST\to TS$ lifts $T$ to a monad on $C^S$. Both are useful in the lifting sense.
If you have a monad and a comonad, say comonad $G$ and monad $S$ then $SG\to GS$ gives simultaneously both lift of $G$ to a comonad on $C^S$ and $S$ to a monad on $C^G$. Namely if $M$ is a $G$-comodule you do composition $SM\to SGM\to GSM$ to obtain a lifted $G$-comodule. If $M$ is an $S$-module you do a composition $SGM\to GSM\to GM$ to get a lifted $S$-module. In that basic sense, in both cases you need a distributive law $SG\to GS$, unlike the two monads case and unlike the two comonads case. This is the reason most references have only one distributive law for mixed case and people in “entwining structure” community repeat that there is only one. I do not understand how to parallel Brookes and Van Stone concept (apart from trivial inversion of the law in the commutative diagrams); how can it be interpreted via any lift ??
P.S. bialgebras in the sense of van Osdol and entwining modules of Brzezinski-Majid are basically the same thing.
P.S.2 Maybe if we interchange EM lifts to Kleisli lifts it would change to the other distributive law. This is again unlike the two monad or two comonads case, but is reasonable as it is 2-categorical setup so it is another duality op versus co.
I am confused now. While the question of lifts above is easily checked, after my crosschecks within old literature (including a paper on entwinings of my own) I need some time to reconcile the two points of view. I remember that Gabi has sent some unexpected dual comonad picture for something similar but different from entwining modules in certain MathReview. I need to compare that and will be back later.
Wisbauer’s survey
states a theorem 5.4 which says roughly what I argued above for the distributive law $SG\to GS$ (citation from above)
If you have a monad and a comonad, say comonad $G$ and monad $S$ then $SG\to GS$ gives simultaneously both lift of $G$ to a comonad on $C^S$ and $S$ to a monad on $C^G$. Namely if $M$ is a $G$-comodule you do composition $SM\to SGM\to GSM$ to obtain a lifted $G$-comodule. If $M$ is an $S$-module you do a composition $SGM\to GSM\to GM$ to get a lifted $S$-module. In that basic sense, in both cases you need a distributive law $SG\to GS$, unlike the two monads case and unlike the two comonads case.
Street in
uses the other version and notices that it is a mate to the distributive laws among two monads (edit: provided the comonad has a right adjoint).
For algebra and a coalgebra one can look at algebra as defining a monad both to the category of left and the category of right modules (edit: over a ground ring). So, in my understanding, for a coalgebra $C$ and algebra $A$, the law $\psi: C\otimes A\to A\otimes C$ can pertain to either of the two kind of laws, as the order is different when we tensor multiply from the left and from the right.
Yes, Power and Watanabe reconcile what was confusing me
Observe that this distributive law allowing one to make a two-sided version of a Kleisli construction is in the opposite direction to that required to build a category of bialgebras.
This explains/acknowledges why the people interested in bialgebras/entwining modules like Brzezinski and me, who look for lifting property and the composed coring see that all 3 constructions (two liftings and the composed coring) require the same direction of the distributive law. I was not aware of this statement about Kleisli construction. (In fact once I complained to Tomasz that the other mixed distributive law seemed to be useful for something and he warned me that there is anyway only one kind for all 3 motivating purposes.)
This corrects my first line in 20 (which is still true if understood in the sense of lifting solution).
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